The central component of a triangulation

TL;DR

This paper introduces the concept of the central component in polygon triangulations, deriving new recurrences for Catalan numbers and enumerating triangulations with fixed central components.

Contribution

It defines the central component in polygon dissections and uses this to establish new recurrences and enumeration formulas for Catalan and k-Catalan numbers.

Findings

Derived new recurrences for Catalan and k-Catalan numbers.

Proved congruence relations for these numbers.

Enumerated triangulations with fixed central components.

Abstract

The central component of a polygon triangulation is defined as the triangle or diameter that contain its geometric center. More generally, every polygon dissection contains a central component. Using this notion, we derive new recurrences for the Catalan and $k$-Catalan numbers, and use these recursions to prove congruence relations of these numbers. We also enumerate the triangulations that contain a fixed vertex in their central components.